To determine the greatest factor that can be factored out of the polynomial [tex]\(30x^5y^8z - 18xy^4\)[/tex], we need to follow these steps:
1. Identify the coefficients and find their greatest common divisor (GCD):
- The coefficients of the terms are 30 and -18.
- The greatest common divisor of 30 and 18 is 6.
2. Determine the variable parts and their exponents:
- For [tex]\(30x^5y^8z\)[/tex], the variables and their exponents are: [tex]\(x^5\)[/tex], [tex]\(y^8\)[/tex], and [tex]\(z^1\)[/tex].
- For [tex]\(-18xy^4\)[/tex], the variables and their exponents are: [tex]\(x^1\)[/tex] and [tex]\(y^4\)[/tex].
3. Identify the common variables and choose the lowest exponent for each:
- The variable [tex]\(x\)[/tex] appears in both terms with exponents 5 and 1. The smaller exponent is [tex]\(x^1\)[/tex].
- The variable [tex]\(y\)[/tex] appears in both terms with exponents 8 and 4. The smaller exponent is [tex]\(y^4\)[/tex].
- The variable [tex]\(z\)[/tex] only appears in the first term, so it is not a common factor between the terms.
4. Combine the GCD of the coefficients and the common variable parts:
- The common factor of the coefficients is 6.
- The common variable parts are [tex]\(x^1\)[/tex] and [tex]\(y^4\)[/tex].
Putting all these together, we conclude the greatest common factor that can be factored out of the polynomial [tex]\(30x^5y^8z - 18xy^4\)[/tex] is:
[tex]\[
6xy^4
\][/tex]
So the correct answer is:
[tex]\[
\boxed{6xy^4}
\][/tex]
